What is an Intuitive Explanation of the Phase of a Signal

Question

I understand that the meaning of the phase response of a system is simply how much the system delays a frequency component. However, I do not find an intuitive explanation for the phase of a signal. The representation the Fourier transform of a signal by polar coordinates, produces two components; the magnitude and the phase. The magnitude response is simply how much each frequency component is contributing to this signal. However, what is the meaning of the phase here? Can I know if a frequency component is "exist" in the signal by just looking into the phase of a signal?

I found an explanation here:

determines how the sinusoids line up relative to one another to form a signal

But the did not help me.

Answer 1

Compare $y= \sin(\omega t)$ to $y=\sin(\omega t-\phi)$ as shown in the plot below. $\phi$ is the additional phase term in radians where $\omega$ represents the frequency in radians per second. Thus the phase term shifts a sinusoid along the horizontal axis.

So at a given frequency this will result in a time delay for that frequency, although to be noted that a fixed time delay will cause a phase shift that is proportional to frequency as a linear variation (The phase will increase linearly as the frequency increases).

Thus our interest in "linear-phase" filters as this will have a constant "group delay" meaning the group of all frequencies in a waveform will be delayed the same amount in time, and thus not cause destructive interference between the different frequencies in the group. (Review Fourier Series and decomposition of waveforms into individual frequencies and the resulting reconstruction for more insight on that).

More pertinent (and useful) is complex signal representation with $Ae^{j \phi}$ representing the magnitude $A$ and phase $\phi$ of a signal. This can be expanded using Euler's identity which may help give further insight: $$Ae^{j\phi}=A\cos(\phi)+jA\sin(\phi) = I + jQ$$

From which we see that in order to completely describe a complex signal we require two real signals, this could be magnitude and phase, $A$, $\phi$ as in $Ae^{j\phi}$ or the real and imagninary components $I$, $Q$ as in $I+jQ$.

Refer to this post which may help give further insight into the importance and utility of complex representation for signals: Frequency shifting of a quadrature mixed signal

Answer 2

The phase of a signal tells one nothing without the magnitude. FFT result bins within a rounding error of zero often have random phases. Whereas the angle of a non-zero length vector actually points somewhere.

Note that a cosine and a sine of the same frequency are orthogonal. A ratio of the two (cosine and sine) is required to represent any sinusoid of that same frequency. The phase tells you the ratio.

Answer 3

The phase of a signal generally refers to the timing of the signal (or how two sinusoids line up) as you posted in your question. But you are asking about the phase of a signal in the frequency domain (i.e., after an FFT operation). The FFT function computes an N-point complex DFT. In practice the real and imaginary components of the result of the FFT operation are used to generate the magnitude spectrum which is usually more useful than the individual I and Q components. The magnitude of the FFT can be used for spectral visualization, energy detection, or a variety of other applications.

Answer 4

Michael Ossman likens the phase of a signal to a slinky (see video here). Instead of thinking of a wave as 2-dimensional, imagine a 3-dimensional helix (a Slinky), where phase is the deviation from the carrier signal.

Answer 5

I make an intuitive understanding of "phase" by using chess.

Wave components with different frequencies correspond to different kinds of chess pieces, and phases are somewhat like piece positions on the chess board. In other words, "phase" represents the structure of a signal, just like the position of each chess piece.

For example, in two chess positions (two waves), both of them have a knight (a sinusoidal component with a specific frequency), but the first one has its knight in the center ("good" phase), the second one has a knight in the corner ("bad" phase). Although both of them have knights, the position of knights are different. The first one has a superior knight, the second one does not. Therefore, the two chess positions are not the same, neither the two waves with different phases of a sine component with the same frequency.

Phase and frequency are two independent quantities (in my knowledge), just like a chess piece and its position. So one cannot infer the existence of a certain frequency from phases.

Moreover, if either magnitude or phase can be preserved to reconstruct the original signal, phase should be selected, not magnitude since phase represents the structure of the frequency representation of the original signal in time domain.

Hope this helps.